Suppose a group of people has to make a choice from a set S of options. Each member of the group ranks the options in S from best to worst. A “voting system” is a mechanism for aggregating these rankings into a single ranking, meant to represent the preferences of the group as a whole.
There are certain natural features you’d like a voting system to have. For instance, you might want it to be “monotone” — if a voter who likes option A better than B switches those two in her ranking, that shouldn’t improve A’s overall position or worsen B’s.
Kenneth Arrow wrote down a modest list of axioms, including monotonicity, that seem like pretty non-negotiable features you’d want a voting system to have. Then he proved that no voting system satisfies all the axioms when S consists of more than two options.
Why wouldn’t that be interesting?
Well, here are some axioms that are not on Arrow’s list:
- Anonymity (the overall outcome is invariant under permutation of voters)
- Neutrality (the overall outcome is invariant under permutation of options)
Surely you don’t really want to consider a voting system that doesn’t meet these requirements. But if you add these two requirements, the resulting special case of Arrow’s theorem was proved more than 150 years earlier, by Condorcet! Namely: it is not hard to check that when |S| = 2, the only anonymous, neutral, Arrovian voting system is majority rule. Add to that Arrow’s axiom of “independence of irrelevant alternatives” and you get
(*) if a majority of the population ranks A above B, then A must finish above B in the final ranking.
But what Condorcet observed is the following discomfiting phenomenon: suppose there are three options, and suppose that the rankings in the population are equally divided between A>B>C, C>A>B, and B>C>A. Then a majority ranks A over B, a majority ranks B over C, and a majority ranks C over A. This contradicts (*).
Given this, my question is: why is Arrow’s theorem considered such a big deal in the theory of social choice? Suppose it were false, and there were a non-anonymous or non-neutral voting mechanism that satisfied Arrow’s other axioms; would there be any serious argument that such a voting system should be adopted?
Thanks to Greg Kuperberg for some helpful explanation about this stuff on Google+. Relevant reading: Ben Webster says Arrow’s Theorem is a scam, but not for the reasons discussed in this post.